Lateral trapping of DNA inside a voltage gated nanopore

The translocation of a short DNA fragment through a nanopore is addressed when the perforated membrane contains an embedded electrode. Accurate numerical solutions of the coupled Poisson, Nernst-Planck, and Stokes equations for a realistic, fully three-dimensional setup as well as analytical approximations for a simplified model are worked out. By applying a suitable voltage to the membrane electrode, the DNA can be forced to preferably traverse the pore either along the pore axis or at a small but finite distance from the pore wall.

Figure 1

Schematic cross section through the considered setup. A cylindrical container of height $H_c$ and radius $R_c$ is filled with electrolyte solution (blue) and is divided by a membrane. The membrane contains a coated gate electrode, a cylindrical nanopore, and an elongated particle (see also Fig. 2 for a closeup). External voltages $V/2$, $-V/2$, and $V_G$ can be applied to the two electrodes at the top and the bottom of the container, and to the membrane gate electrode, respectively. The origin of the Cartesian coordinates is located at the pore center, but is drawn here off the center for better visibility. The intersection with the $x_1=0$ plane is indicated as thin black line to support the perspective view. The entire sketch is drawn to scale, adopting the standard values from Sec. 2a which will be used for the numerical solutions in Sec. 5.

Figure 2

Closeup of the pore region from Fig. 1 (again drawn to scale) with pore radius $R_p$, gate electrode (gray) of thickness $T_e$, coating (yellow), and total membrane thickness $T_m$. The particle (green) of radius $R$, length $L$, and distance $r$ from the pore axis (black line) may model, e.g., a DNA segment or an elongated nanoparticle.

Figure 3

Sketch of the simplified setup considered in Sec. 3. A planar gate electrode at $x=0$ is subject to a gate voltage $V_G$ relatively to the reference value $V=0$ at $x=\infty$. The electrode is coated by an infinitely thin, dielectric (nonconducting) layer with permittivity ${\epsilon }_m$ and surface charge density ${\sigma }_m$ (omitted in the figure). At a distance $d$ there is a dielectric plate of thickness $2R$ parallel to the electrode (region II, permittivity ${\epsilon }_p$). The gap between electrode and plate (region I) as well as the region beyond the plate (region III) is filled with electrolyte solution (permittivity ${\epsilon }_s$).

Figure 4

Potential energy per area from (20) versus $r$ from (21) for the effectively one-dimensional setup from Fig. 3, six different $V_G$ values as indicated, ${\sigma }_p=-0.01\mathrm{C}/{\mathrm{m}}^2$, $c_0=100\mathrm{mM}$, and all the remaining parameters as specified in Sec. 2a. The energy is given in units of thermal energy $k_{B}T$ (at room temperature) per surface area (in ${\mathrm{nm}}^2$) of the dielectric plate in Fig. 3. Lines: numerically exact solutions. Symbols: analytical approximation from (22). As detailed in the main text, the variable $r$ from (21) is adopted for the sake of better comparability with the setups considered later on. Since $R=1.1\mathrm{nm}$ and $R_p=5\mathrm{nm}$ (see Sec. 2a), $r$ is restricted to values $\le 3.9\mathrm{nm}$, and $r=3.9\mathrm{nm}$ corresponds to $d=0$, i.e., to the situation where the dielectric plate touches the gate electrode in Fig. 3. The results for $r<0$ (i.e., large $d$) are of little interest and have been omitted. The transition from a monotonic to a nonmonotonic behavior of $U\left(d\right)$ takes place close to the depicted cases with $V_G=-0.0136\mathrm{V}$ and $V_G=0.0143\mathrm{V}$.

Figure 5

Same as in Fig. 4 but with ${\sigma }_p=-0.05\mathrm{C}/{\mathrm{m}}^2$ (i.e., the standard surface charge density of our DNA model, see Sec. 2a) and modified gate voltages $V_G$.

Figure 6

Cross section through the effectively two-dimensional model, consisting of an infinitely long, cylindrical pore and an infinitely long, rod-shaped DNA of radius $R$ parallel to the pore axis. Similarly as in Figs. 1 and 2, the electrolyte solution (blue) is confined by a cylindrical gate electrode (gray) with gate voltage $V_G$ and with a dielectric coating (yellow) of thickness $C$, inner radius $R_p$, and surface charge density ${\sigma }_m$.

Figure 7

Potential energy per unit length versus $r$ for the effectively two-dimensional model from Sec. 4a (see also Fig. 6) with $C=0$ (infinitely thin coating) and pore radius $R_p=5\mathrm{nm}$. All other parameter values are as in Fig. 4 (except that two $V_G$ values have been omitted in order not to overload the figure). Lines: numerical solutions. Symbols: same analytical approximation (22) as in Fig. 4, but multiplied by $2.2\mathrm{nm}$ to roughly account for the “effective DNA width” in the one-dimensional model (see Sec. 3b).

Figure 8

Same as Fig. 7 except that the parameter values are chosen as in Fig. 5. The additional dashed lines for $V_G=0.15\mathrm{V}$ and $V_G=-0.1\mathrm{V}$ are the numerical solutions from Fig. 5 multiplied by $2.2\mathrm{nm}$.

Figure 9

Potential energy per unit length versus $r$ for the effectively two-dimensional model from Sec. 4a with pore radius $R_p=5\mathrm{nm}$, coating thickness $C=2\mathrm{nm}$ (see Fig. 6), surface charge densities ${\sigma }_m=0.01\mathrm{C}/{\mathrm{m}}^2$, and ${\sigma }_p=-0.05\mathrm{C}/{\mathrm{m}}^2$ (see Sec. 2a), and bulk concentration $c_0=10\mathrm{mM}$. All other parameter values are as specified in Sec. 2a.

Figure 10

Potential energy $U\left(r\right)$ in units of the thermal energy $k_{B}T$ (at room temperature) for two different gate voltages $V_G$. Solid line: numerical results for the fully three-dimensional setup, obtained from (1) as detailed in the main text. Dashed line: same as the corresponding two curves in Fig. 9, but multiplied by the pore length $T_m=15\mathrm{nm}$ to approximately account for the “effective system length” in the two-dimensional model from Sec. 4a.

Figure 11

Representative numerical solution of the coupled PNPS problem from Sec. 2b for the setup from Figs. 1 and 2 and with the standard parameter values as specified in Sec. 2a, external voltage $V=50\mathrm{mV}$, gate voltage $V_G=300\mathrm{mV}$, and bulk concentration $c_0=10\mathrm{mM}$. Depicted is the color coded pressure (in Pa) on the surface of the porous membrane and the rod shaped DNA. More precisely, we plot the excess pressure over the preset value (zero) at the external electrodes in Fig. 1.